ASYMPTOTIC BEHAVIOR OF A SECOND-ORDER SWARM SPHERE MODEL AND ITS KINETIC LIMIT

. We study the asymptotic behavior of a second-order swarm model on the unit sphere in both particle and kinetic regimes for the identical cases. For the emergent behaviors of the particle model, we show that a solution to the particle system with identical oscillators always converge to the equilibrium by employing the gradient-like ﬂow approach. Moreover, we establish the uniform-in-time (cid:96) 2 -stability using the complete aggregation estimate. By applying such uniform stability result, we can perform a rigorous mean-ﬁeld limit, which is valid for all time, to derive the Vlasov-type kinetic equation on the phase space. For the proposed kinetic equation, we present the global existence of measure-valued solutions and emergent behaviors. and discuss the basic


1.
Introduction. Emergence of collective behavior has been widely studied not only in applied mathematics, but also in other scientific disciplines, for instance, control theory in engineering community [24,33,36], active matter in statistical physics [3,28,31] and swarming behavior in quantitive biology [14,17,34]. In spite of its crucial role in biological processes, it has been only fifty years since the mathematical study of such collective motion started after seminal work of Winfree [38] and Kuramoto [25]. Among well-known models describing collective oscillatory behavior, to name a few, the Cucker-Smale model [13], the Kuramoto model [25] and the Vicsek model [37], our main interest lies in the emergent dynamics on the d-dimensional unit sphere S d−1 embedded in the d-dimensional Euclidean space R d . For the simplest case, the first-order dynamics has been used to describe aggregation of particles whose norms are preserved for all time: Here, ·, · and | · | denote the standard inner product in R d and its induced norm in R d , respectively, Ω i ∈ Skew d (R) is a d×d skew-symmetric matrix so that each norm of x i is conserved along the dynamics, and κ stands for a positive coupling strength.
In addition, the communication term in the right-hand side can be interpreted in 402 DOHYUN KIM the following way [32]: each agent moves towards the projected average position of its neighbors on a sphere. See the last paragraph in Section 2.1 for literature review for (1).
Recently, Ha and Kim proposed a second-order extension to the first-order model (1) in [18] by incorporating an inertial force and centripetal force: where m and γ are nonnegative mass and friction coefficient, respectively. For the newly proposed model (2), it can be heuristically derived from Hamiltonian dynamics theory together with a frictional force, an one-body force field and an interaction force field. For more detailed derivation, we refer the reader to Appendix A in [35] or Section 2.1 in [18]. Same as the first-order model (1), we see that the unit sphere is positively invariant under the flow (2) (see Lemma 2.2). Then, (2) can be reduced to the model on S d−1 × R d : In [18], they provided some restricted classes of initial data leading to the complete aggregated state and incoherent state for attractive coupling and repulsive coupling regimes, respectively. The main result of this paper consists of two parts: the particle model (3) and its kinetic model (4). Our first main result concerns with asymptotic behavior for the particle model. More precisely, we utilize the gradient-like flow approach to show that a solution to (3) always converges to the equilibrium (see Theorem 3.1). However, this does not imply that the complete aggregation occurs for all initial data (see Definition 2.1 for the definition of the complete aggregation). Together with the linear stability analysis performed in Section 4.2 of [18], we would say that the complete aggregation emerges for generic initial data. Note that the gradient-like flow approach to the second-order system is first studied in [23] and after then, this work has been applied in the context of collective dynamics. See for instance [11,12] in which the gradient-like flow approach is used for the inertial Kuramoto model. On the other hand, uniform-in-time 2 -stability with respect to the initial data is established in which we a priori assume that the complete aggregation occurs. This kind of stability theorem states that difference of any two solutions can be uniformly controlled by difference of initial data. To be more specific, for any two solutions Z andZ to (3), there exists a uniform constant C which does not depend on N and t such that sup 0≤t<∞ |Z(t) −Z(t)| ≤ C|Z(0) −Z(0)|.
It follows from the literature that finite-in-time estimate always holds when the system admits a sufficiently regular solution, in other words, for any T > 0, the uniform stability estimate holds until such a time, that is, it holds on [0, T ). However, T does not allow infinity in general. To investigate asymptotic behavior of a solution towards t → ∞, we need T = ∞. In fact, in Theorem 3.3, we show that the uniform 2 -stability is valid for all time, hence T can achieve infinity.
Our second main result deals with rigorous derivation of a kinetic description for (3) using uniform-in-time 2 -stability. As a result, we obtain the kinetic equation for a density f = f (t, ω, v, Ω) corresponding to (3): where the interaction term is defined as Indeed, by following the same procedure of the first-order model, we first identify the kinetic equation for (2) defined on R d × R d × Skew d (R) and then it follows from the invariance of manifold D that our target equation (4) is derived from the method of characteristic (see Section 2.3). Once we attain the kinetic equation from the particle system using mean-field limit, then the global existence of measure-valued solution for the kinetic equation directly follows by measure-theoretical formulation (see Theorem 3.4). For detailed description, we refer the reader to Section 5 of [22] for the measure-valued solutions for the Cucker-Smale dynamics. Moreover, asymptotic behavior of the measure-valued solutions can be obtained by using the method of characteristic and by handling the particle and kinetic models in a common framework. Thus, it suffices to lift the results for the particle model to the kinetic model (see [9] for a parallel story of the inertial Kuramoto model). For this, we derive the energy estimate to show that the kinetic energy converges zero with an integrable decay rate. In addition, we see that the measure-valued solution tends to a bi-polar state (see Theorem 3.5 and Corollary 1).
The rest of this paper is organized as follows. In Section 2, we briefly introduce recent progress on the first-order swarm sphere model and present previous results on (3). Moreover, we discuss the basic properties of our main model (3) and its kinetic model (4). Section 3 contains the summary of the main results whose proofs will be completed in the following two sections. In Section 4, we first study the emergent behavior of the particle model in which the gradient-like flow approach and uniform-in-time 2 -stability are used. Section 5 deals with the kinetic model derived from the particle model and the global existence of a measure-valued solution. Moreover, we present the asymptotic behavior of the measure-valued solution using energy estimate. Finally, Section 6 is devoted to a brief summary of our main results and remaining issues for future work.
Notation. For vectors x = (x 1 , · · · , x d ) and y = (y 1 , · · · , y d ) in R d , we set the p -norm and inner product ·, · in R d : In particular, we denote |x| := x 2 = x, x for the simplicity.

2.
Preliminaries. In this section, we begin with a brief story of the first-order swarm sphere model which gives an outline of the paper, and discuss the basic properties of the particle model (3) and the kinetic model (4). Moreover, we present the previous results of (3) and introduce some preparatory lemmas for later use, and provide the relation between (3) and the other models for collective dynamics such as the inertial Kuramoto model and the first-order swarm sphere model.

2.1.
A swarm sphere model. Here, we report recent progress on the first-order swarm sphere model which generalizes the well-known Kuramoto model into highdimensional space: for Here, Ω i ∈ Skew d (R) is a d × d skew-symmetric matrix so that each norm of x i is conserved along time, and κ denotes a positive coupling strength. Hence, if we choose the initial data satisfying x 0 i ∈ S d−1 , that is, |x 0 i | = 1 for all i = 1, · · · , N , then system (1) reduces to the model on S d−1 : In fact, the communication term in the right-hand side is represented as the orthogonal projection onto the tangent plane [21,30]. Hence, one can rewrite (2) as the gradient flow (see Section 4.1). We here recall the definition of complete aggregation.
Then, we say that system (3) or (2) exhibits the complete aggregation, if the following relation holds: On the other hand, in [19], the continuum equation for the one-particle distribution function F = F (t, x, Ω) corresponding to (1) is formally derived using standard BBGKY hierarchy: As in the particle model, we also check that the domain E := S d−1 × Skew d (R) is invariant so that continuum equation (3) can be reduced to the following equation defined on E: , Moreover, when the identical oscillators are considered, that is, Ω i ≡ Ω for i = 1, · · · , N , rigorous uniform-in-time derivation of (4) from (2) is obtained in [19].
To the best of our knowledge, uniform-in-time mean-field limit is first presented in [20] where uniform p,q -stability estimate for the Cucker-Smale flocking model is established.
The first-order model (2) and (4) have been extensively studied by several groups, to name a few, Frouvelle and Liu [16], Ha [7,19,21], Lohe [26,27], Markdahl [29,30], Olfati-Saber [32], Ott [5,6], Piccoli [4] and Zhu [39,40]. Below, we present brief literature review. For the particle model (2) with identical oscillators, emergence of the complete aggregation has been considered in different contexts. In [32], they first classified the equilibrium of (2) with Ω i ≡ O into three types: completely aggregated state, bi-polar state and dispersed (or incoherent) state. Then, they performed a (formal) linear stability analysis of the equilibrium for the system to see that the last two equilibria are unstable. Note that (2) has been also studied in the opinion dynamics [1,4]. In [4], they employed a specific network structure and a control term into the model to study how they affect the asymptotic behavior of (2). See also [1] for the opinion dynamics on the compact Riemannian manifold. In [39], they constructed a potential function associated to (2) and applied LaSalle's invariance principle to verify that all vectors {x i } N i=1 converges to the same point. In [29], they formulated system (2) with Ω i ≡ O as the gradient flow and use Lojasiewicz inequality to show that the complete aggregation occurs for generic initial data. In [21], they obtained the similar results as shown in [29] simultaneously, and provided a dichotomy for all initial data. More precisely, if ρ(0) = 0, then ρ(t) ≡ 0 for all time. Otherwise, if ρ(0) > 0, only two cases are possible: completely aggregated state and bi-polar state with only one antipodal position. For the latter case, they showed that there exists one and only one single particle that converges to the point, say x * , and the remained N − 1 particles converge to the antipodal points −x * . On the other hand for the continuum model (4), in [19], its rigorous and uniform-intime derivation of (4) is obtained using uniform stability estimate. In [16], they studied the asymptotic behavior of a measure-valued solution to (4) and showed that the measure-valued solution tends to a bi-polar state. In [5], they observed that discontinuous phase transition occurs nonhysteretically in odd dimension d ≥ 3, whereas only continuous phase transition can arise in the Kuramoto model.

Basic properties.
In this subsection, we study the basic properties of (3), such as the invariance of underlying manifolds, orthogonal invariance and solution splitting property.
Lemma 2.2. Let X = (x 1 , · · · , x N ) be a solution to (2) satisfying the initial data: Then, the modulus of x i is a constant of motion.

DOHYUN KIM
Thus, one has Although its proof can be found in Lemma 2.1 of [18], we provide its proof for the consistency of the paper. We take an inner product (2) with x i to obtain For the handy notation, we set By differentiating u i , one haṡ Then, the initial condition (5) is rewritten aṡ We also rewrite (6) in terms of u i : Hence, in (8), we use the initial condition (7) to obtain the desired result: Due to Lemma 2.2, system (2) can be written as the simplified system (3) defined on S d−1 × R d : Next, we show that system (3) with Ω i ≡ O has the rotational symmetry. For a d × d orthogonal matrix U , we defined transformed variables: Lemma 2.3. Let X = (x 1 , · · · , x N ) be a solution to (3) and U be a d×d orthogonal matrix. Then, the transformed variableX = (x 1 , · · · ,x N ) is also a solution to (3) with Thus, for Ω i ≡ O, we multiply (3) with U to obtain Finally, we use the notation (9) to show the desired assertion.
Remark 1. (i) In fact, the orthogonal invariance property does not hold in general for system (3) with distinct Ω i .
(ii) It is worthwhile to mention that system (3) itself does not satisfy the solution splitting property as it is. In order for the property to be valid, we have to modify (3) as follows: Note that the last two terms are added to preserve the solution splitting property. More precisely, we set y i := e − Ω γ t x i . Then, its derivatives becomė Since e − Ω γ t is an orthogonal matrix, we can use the property (10). For Ω i ≡ Ω, we multiply (11) with e − Ω γ t to find We use (12) to represent (13) in terms of y i : which can be rearranged as After tedious calculation, one can check that K vanishes, that is, K = 0. Thus, (11) admits the desired solution splitting property. We summarize the discussion above in the following lemma. (11) and suppose that Then, y i := e − Ω γ t x i is a solution to the following reduced equation: 2.3. From particle swarm to kinetic swarm. We investigate a kinetic model which can be naturally derived from (2) using standard BBGKY hierarchy. After performing formal mean-field limit, we consequently identify the desired equation Lemma 2.2 yields that the d-dimensional unit sphere S d−1 is positively invariant under the flow (2). By a similar fashion, we show that our domain D = S d−1 × R d ×Skew d (R) is a positively invariant manifold for the kinetic model (14). In order to distinguish the notation for the densities, we set f := F | D to derive our kinetic model on the unit sphere.
x, v, Ω) be a continuously differentiable solution to (14) so that the calculation below can be performed. Then, the following two assertions hold.
Proof. First, we rewrite (14) into a quasilinear form: For a given x ∈ R d and Ω ∈ B ν ∞ (0), we set a forward characteristic associated to (2) (X(s), V (s), Ω(s)) := (X(s; Then, the first assertion directly follows from the relation (15) 3 . For the second assertion, we differentiate the term X(s), V (s) to observe Here, we used the following identity in the third equality: Since we initially assume that the uniqueness of a solution to (16) yields the desired positive invariance of D.
Thanks to Lemma 2.5, the equation (14) for From now on, we only consider the equation (4) for f :

Previous results.
In this subsection, we briefly review the previous results. First, we present the asymptotic behavior of the velocity variableẋ i . For a solution X to (1), we define the Lyapunov functional measuring the degree of aggregation: Then, we can derive the dynamics of L + and arrive at the following result. For the proof, we refer the reader to Remark 4.1 and Theorem 4.4 of [18]. Proposition 1.
[18] Suppose that the system parameters and the initial data satisfy m > 0, γ > 0, κ > 0, L 0 + < ∞, and let X be a solution to (1). Then, there exists a function

DOHYUN KIM
Hence, we have the zero convergences of the velocity variables with an integrable decay rate: Next, we consider the asymptotic behavior for the position variable x i . We observe the identity: Hence, it suffices to focus on the dynamics of relative angles h ij := x i , x j . For the evolution of them, we introduce two frameworks (large mκ and small mκ regimes) leading the complete aggregation. Let δ ∈ (0, 1) be a fixed (small) positive number and G = G(t) be the maximal diameter for 1 − h ij : •(F A 2): There exists small numbers ε k A , k = 1, 2, 3 depending only on the system parameters such that Framework (F B ) (small mκ regime): •(F B 1): Parameters m, γ, κ and δ satisfy •(F B 2): There exists small numbers ε k B , k = 1, 2, 3 depending only on the system parameters such that For exact values of the constants ε k A and ε k B , we refer the reader to Section 4.3 in [18]. Under exactly one of these frameworks (F A ) and (F B ), we show that the complete aggregation occurs. We refer the reader to Theorem 4.10 in [18] for the proof.
Theorem 2.6. [18] Suppose that exactly one of frameworks (F A ) and (F B ) holds, and let X be a solution to (1).
Hence, the system exhibits the complete aggregation with an integrable decay rate, in other words, lim t→∞ G(t) = 0.

Preparatory lemmas.
For later use, we recall Barbalat's lemma and Grönwalltype lemmas. (ii) If f is continuously differentiable, lim t→∞ f (t) ∈ R, and f is uniformly continuous, then f tends to zero as t → ∞: is an integrable and bounded function, and a, b and c are positive constants. Let y = y(t) be a nonnegative C 2 -function satisfying exactly one of the following differential inequalities: Then, in any cases, we have y ∈ (L 1 ∩ L ∞ )(0, ∞). In other words, lim t→∞ y(t) = 0 with an integrable decay rate.
Proof. Convergence to zero of y in both cases can be found in Lemma A.1 for [8] and Lemma 4.9 for [18], respectively. We here discuss the integrability of y. For the first assertion, it follows from [8] that Since the exponential function and f are integrable, we see that y is also integrable. For the second assertion, we consider the two cases: b 2 − 4ac < 0 and b 2 − 4ac > 0. If b 2 − 4ac < 0, then there exist positive constants C k n , k = 1, · · · , 4 such that Thus, integrability of y follows from the integrability of e −t , te −t and f . On the other hand, if b 2 − 4ac > 0, then there exist positive constants C k p , k = 1, · · · , 4 such thatẏ Then, since the right-hand side of (19) is integrable, one applies (18) to obtain the desired result.
Before we end this section, we briefly discuss the relation between (3) with other two collective models, namely the inertial Kuramoto model [10,15] and the firstorder swarm sphere model (1). First, we present the connection with the inertial 412 DOHYUN KIM Kuramoto model. Let x i ∈ R 2 be a solution to (3) with |x i | = 1. Then, we can represent x i in terms of a polar coordinate: Then, we observeẋ By substituting the ansatz (20) and (21) into (3), one has Now, we multiply the above relation by e −iθi to see We finally compare the imaginary parts of the both sides of (22) to obtain the inertial Kuramoto model: On the other hand, if we formally set m = 0, then (3) reduces to the first-order model:ẋ Thus, we would say that our model (3) is a second-order extension of (23).
3. Description of main results. In this section, we discuss our main results without detailed proofs, which will be presented in the following two sections, and the main results can be divided into two parts: the particle model in Section 3.1 and the kinetic one in Section 3.2. In what follows, unless otherwise stated, we mainly deal with the identical case, and due to the solution splitting property, we may assume that Ω i ≡ O so that the identical particle model becomes and hence its corresponding kinetic model reads as follows: for f = f (t, ω, v), However, if we divide (1) by m, then γ/m and κ/m play the same roles as γ and κ, respectively. Thus, without loss of generality, we henceforth set m = 1 in both (1) and (2) to avoid possible cluttered estimates.
3.1. Particle model. First, we provide gradient-like flow formulation of (3). Since our system contains the centripetal force term |ẋ i | 2 x i , the classical theorem (Theorem 4.1) in [22] could not be applied as it is. Hence, we follow the proof of the theorem and slightly modify it to show our first main result.
) be a solution tö Suppose that g and F satisfy the structural condition: there exist positive constants Proof. The detailed justification will be given in Section 4.1.
Before we state our second main result, we introduce the definition of the uniform 2 -stability with respect to the initial data. Recall | · | denotes the usual 2 -norm in the Euclidean space.
Definition 3.2. We say that system (1) is uniformly 2 -stable with respect to the initial data if for any two solutions Z andZ with initial data Z 0 andZ 0 , respectively, there exists a uniform positive constant C which does not depend on N and t such that sup Remark 2. If the maximal lifespan of time satisfying the estimate (5) is finite, then we say that the estimate is local-in-time or system (1) is local-in-time stable.
Then, we find the desired uniform constant G when we a priori assume that the complete aggregation occurs. Theorem 3.3. Suppose that the damping coefficient is larger than the inertia assumed to be 1, that is, γ > 1, (33) and that the framework (F A ) or (F B ) holds so that the emergence of the complete aggregation is guaranteed. Let Z andZ be two solutions to (1) with initial data Z 0 andZ 0 , respectively. Then, there exists a uniform positive constant G satisfying (5). In other words, system (1) is uniformly 2 -stable in the sense of Definition 3.2.
Proof. The detailed proof can be found in Section 4.2.
3.2. Kinetic model. It follows from the uniform-in-time 2 -stability in Theorem 3.3 that global existence and uniform-in-time stability of a measure-valued solution to (2) directly follow. For a metric space X , we denote P 2 (X ) as the set of Borel probability measures on E with finite moments of order two.
(ii) (Uniform stability): If f andf are two solutions to (2) with the initial data f 0 andf 0 , respectively, then there exists a uniform constant G independent of t such that Proof. We postpone the rigorous justification in Section 5.1.
We finally state our last main result which concerns with the asymptotic behavior of a measure-valued solution whose global existence is guaranteed from Theorem 3.4. For this, we consider the characteristic system associated to (2) with identical oscillators: and define the order parameter Theorem 3.5. Suppose that the initial data satisfy and let f ∈ C w (R + ; P 2 (D)) be a measure-valued solution to (2). Then, the following assertions hold: (ii) If R 0 f > 0, then the measure-valued solution dµ t = f (t)dS ω dv tends to the bipolar state. More precisely, there exists a constant m 0 ∈ (0, 1) and a vector u ∈ S d−1 such that f tends to where δ denotes the Dirac measure.
Proof. We provide the proof in Section 5.2.
4. Asymptotic behavior of the particle swarm sphere model. In this section, we study the asymptotic behavior of the identical second-order particle swarm sphere model (1), and present the proofs of Theorems 3.1 and 3.3.

Proof of Theorem 3.1.
In this subsection, we provide the proof of Theorem 3.1 which deals with the gradient-like flow approach. In [21,30], a first-order swarm sphere model (1) can be represented as a gradient flow only if the natural frequencies are identical to the zero matrix, i.e., Ω i ≡ O. More precisely, we define an analytic potential function V associated to a solution for (1): Then, one can rewrite (1) with Ω i ≡ O aṡ Note that the right-hand side denotes the orthogonal projection of the gradient vector ∇ xi V ∈ R d onto the tangent plane T xi S d−1 at x i , which becomes the induced gradient vector on the manifold S d−1 . It is worthwhile to mention that the orthogonal projection of the usual gradient onto the tangent plane defines the Riemannian gradient. To be more specific, for a Riemannian submanifold M of R d , letf be a function defined on R d and f be the restriction off to M, that is, f =f M . Then, the Riemannian gradient can be calculated as where grad denotes the Riemannian gradient and Proj x : R d → T x M is the orthogonal projection onto the tangent space at x ∈ M. In a similar fashion to (1), we rewrite (1) a gradient-like system: Before we provide the proof of Theorem 3.1, we recall the classical theorem in [23] which concerns with the convergence of a solution to the gradient-like system.
Suppose that g and F satisfy the conditions (4). Denote the set of critical points of the gradient vector field ∇F : Then, there exists a ∈ S such that lim t→∞ |Ẏ (t)| + |Y (t) − a| = 0.
In fact, we notice that Theorem 4.1 is formulated on the Euclidean space R d , whereas our system (1) is defined on the unit sphere. Moreover, system (1) cannot be written as the form of (3). In other words, since (2) contains the centripetal force term |ẋ i | 2 x i so that it cannot be a function of onlyẋ i , we cannot associate such a function g depending only onẋ i as in Theorem 4.1. Thus, we cannot apply Theorem 4.1 as it is. To overcome the technical problems, we follow the proof of Theorem 4.1 step by step to recover the same result. For this, we recall the Lojasiewicz inequality in a Riemannian mainfold whose proof can be found in Theorem 5.1 of [21].
Together with Theorems 4.1 and 4.2, we now present the proof of Theorem 3.1.
(Proof of Theorem 3.1) Our proof consists of two steps: convergences ofẏ i and y i .
• Step A (convergence ofẏ i ): we take an inner product (3) withẏ i and integrate the resulting relation with respect to time to find Then, we recall (4)(i) to see Since F is analytic, we have |ẏ i | 2 ∈ L 1 (0, ∞). On the other hand, since y i andẏ i are uniformly bounded, it follows from (3) thatÿ i is also bounded. This implies that |ẏ i | 2 is uniformly bounded and hence uniformly continuous. Thus, Barbalat's lemma yields the first desired convergence: • Step B (convergence of y i ): For a given initial data (y 0 i ,ẏ 0 i ), we define an ω-limit set: ω(y 0 i ,ẏ 0 i ) := {u ∈ S d−1 : ∃t n → ∞ such that y i (t n ) → u}. Then, since S d−1 is compact, the ω-limit set ω(y 0 i ,ẏ 0 i ) is also a nonempty compact set and hence a connected set. In addition, if we use the definition of the ω-limit set, the governing equation (3) and the convergence (5), then one can check that To verify the desired convergence, we use the analytic condition on F . First, without loss of generality, we can set . For a small positive number ε ∈ (0, 1) which will be determined later in (6) and (8), we associate the energy functional to the system We differentiate the functional E i to obtaiṅ where ∇ 2 F (y i ) denotes the Hessian matrix of F . We choose ε to satisfy where · ∞ denotes the maximum norm of a given matrix. Then, we observė where we used the uniform bound, say M , of max iẏi in the second inequality and the young's inequality is used in the third inequality. Moreover, the positive constant ν is defined as For the negative sign of the right-hand side for (7) or the positivity for ν, we choose δ and ε to satisfy Thus, we findĖ Since ω(y 0 i ,ẏ 0 i ) is a nonempty compact set, we choose y ∞ i ∈ ω(y 0 i ,ẏ 0 i ). Hence, we attain lim and it follows from the Cauchy-Schwarz inequality that Since y ∞ i ∈ ω(y 0 i ,ẏ 0 i ), there exists a sequence (t n ) n≥1 such that lim n→∞ y i (t n ) = y ∞ i .

DOHYUN KIM
Thus, for arbitrary small σ > 0 and C 0 := max{2, 1 + C L }, there exists a large number N 0 such that for n ≥ N 0 , Define a time T * as From the definition of T * , one has lim t→T * In (11), we use Theorem 4.2 and (12)(iii) to obtain Now, we consider two cases: first, if there exists t 0 ∈ R + such that Then, the relation (9) yields E(t) = 0 for t ≥ t 0 and this shows that our solution converges to a stationary state. Otherwise, in (10), we combine (7) and (14) to obtain We integrate (15) over the interval (t N0 , T * ) to find Now, suppose to the contrary that T * < ∞. Then, we use (12) and (16) to see which contradicts (13). Hence, T * = ∞. Finally, if we invoke (16), then we can show that the limit of y i exists: This completes the proof.

Proof of Theorem 3.3. In this subsection, we establish the uniform-in-time
2 -stability with respect to the initial data of the second-order particle swarm model (1) in which the proof of Theorem 3.3 is provided. For this, we rewrite the system into the first-order dynamics introducing the velocity variable v i :=ẋ i . Then, our system reads as Let (X, V ) and (X,Ṽ ) be two solutions to (17). For notational simplicity, we set For the moment, we here use the notation · for vectors in R dN to distinguish the notation | · | for those in R d . Then, we observe Below, we present estimates of I 21 and I 22 , respectively.
(Estimate of I 21 ) We recall the relation in Lemma 2.2: For handy notation, we set and it follows from Proposition 1 that   Then, we estimate I 21 as follows: (Estimate of I 22 ) Note that the following relations hold: Hence, the term I 22 can be calculated in the following way: We recall the complete aggregation estimate in Theorem 2.6: Then, we can further estimate I 22 in (20): In (18), we combine (19) and (22) to obtain We observe and it follows from the Cauchy-Schwarz inequality that (59) Hence, we sum (23) over all the index i = 1, · · · , N and use (24)-(25) to find We summarize the previous estimates in the following lemma. Let (X, V ) and (X,Ṽ ) be two solutions to (17). Then, we have We now consider the time-evolution of X −X, V −Ṽ . For this, we first see Below, we present estimates of I 23 and I 24 , separately.
• (Estimate of I 23 ): We use the following identity: to calculate I 23 : • (Estimate of I 24 ): We observe In (27), we collect (28)- (29) and multiply the resulting relation by ε 1 ∈ (0, 1) to find Then, by summing (30) with respect to i = 1, · · · , N and using the relations (21) and (25) to obtain Our estimate for X −X, V −Ṽ can be stated in the following lemma.
Lemma 4.4. Let (X, V ) and (X,Ṽ ) be two solutions to (17). Then, we have Now, we add (26) and (31) to yield d dt We finally prove Theorem 3.3 applying Lemmas 4.3 and 4.4.
(Proof of Theorem 3.3) To avoid the cluttered mathematical expressions, we rewrite (32): and then provide estimates of I 25 and I 26 , respectively.
• (Estimate of I 25 ): It follows from the Cauchy-Schwarz inequality that Then, I 25 can be estimated as follows: where the constant β > 0 is defined by Note that since we assume and ε 1 ∈ (0, 1) the condition (6), β is positive.
• (Estimate of I 26 ): We observe where the function J = J (t) is defined through the relation: Then, it follows from Proposition 1 and Theorem 2.6 that J is integrable: We integrate (33) to see We recall the definition of I 25 to find Finally, we apply the Gröwnall's lemma to find our desired uniform constant: where the positive constant C is defined as This establishes the proof.

5.
Asymptotic behavior of the kinetic swarm sphere model. In this section, we study the asymptotic behavior of a measure-valued solution to the kinetic swarm sphere model (2), and present the proof of Theorems 3.4 and 3.5.
5.1. Proof of Theorem 3.4. In this subsection, we present the proof of Theorem 3.4. For this, we provide a measure-theoretic preliminaries for our discussion. First, we recall the definition of the measure-valued solution to (4).
) is a measure-valued solution with initial measure µ 0 ∈ P 2 (D), if the following three assertions hold.
(ii) µ is weakly continuous in t: ). (iii) µ satisfies the kinetic equation (4) in the following weak sense: for all test Here, we adopt a standard duality relation: for f ∈ C 0 (D) and µ ∈ P 2 (D), Below, as mentioned in Section 1, we see that (3) and (4) can be treated in a common setting. For any solution (ω i , v i ) to (17), we associate the empirical measure Then, if the empirical measure f N is acted on (4), then one can recover (3). This allows us to deal with (3) and (4) in the same framework. We now determine how to define the distance between two probability measures. Among many candidates, we adopt Wasserstein-2 distance, denoted as W 2 , in the space P 2 (D), the set of Borel probability measures on D with finite moments of order two.
Definition 5.2. (i) For two measures µ, ν ∈ P 2 (D), we define the Wasserstein-2 distance between µ and ν as where Γ(µ, ν) represents the collection of all measures in D × D with marginals µ and ν.
(ii) The kinetic equation (4) is derivable in [0, T ) from the particle level (3) if the following properties hold.

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(A): For a given initial measure µ 0 ∈ P 2 (D) to (4), µ 0 can be approximated by the initial empirical measure µ N 0 corresponding to (3): lim (B): For t ∈ [0, T ), there exists a unique measure-valued solution µ t of (4) with initial data µ 0 , and such solution µ t can be approximated by the empirical measure ν N t of (3) uniformly in time interval [0, T ), where µ N t is the measure-valued solution of the particle system (3): Hence, it follows from the triangle inequality and (2) that Remark 3. Note that the previous uniform stability result in Theorem 3.3 crucially depend on the condition: which guarantees the emergence of the complete aggregation. However, if we do not impose the condition (4), then we would not achieve uniform-in-time results. For instance, consider the general non-identical case, that is, Ω i = Ω j for some i = j. Then, the estimate (5) in Theorem 3.3 can be stated in the following way: Thus, the measure-valued solution to kinetic equation (2) exists globally but not globally-in-time. In other words, global-in-time existence of a measure-valued solution is possible for (2), however, it would not be possible for (4).

5.2.
Proof of Theorem 3.5. In this subsection, we provide the proof of Theorem 3.5 which concerns with the asymptotic behavior of the kinetic equation (2). For our estimate, we a priori assume that our measure-valued solution f ∈ C w (R + ; P 2 (D)).
For the notational simplicity, we set Then, we can write

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As in the particle level, we define the energy functional to (2) which is reminiscent of (17): where E K and E P denote kinetic and potential energies. Below, we present the energy estimate of the kinetic equation (5).
(ii) By differentiating the kinetic energy, one has On the other hand, we find Finally, we substitute (10) into (9) to find the desired energy relation: As a direct consequence of Lemma 5.3(ii), we see that the kinetic energy tends to zero.

Corollary 1.
Let dµ t = f (t)dS ω dv ∈ C w (R + ; P 2 (D)) be a measure-valued solution to (5). Then, the kinetic energy converges to zero with unknown but integrable decay rate: Proof. By integrating the relation in Lemma 5.3(ii), we find

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In order to apply Barbalat's lemma, we need to check that E K is uniformly continuous. For this, we use Hölder's inequality to see In (9), we use (12) together with the notation y := E Hence, we find the uniform upper bound for E K : In addition,Ė K is uniformly bounded: Thus, we verify that E K is uniformly continuous so that the relation (11) yields the desired convergence: lim t→∞ E K (t) = 0. Lemma 5.3 and Corollary 1 concern with the evolution of the velocity variable. In what follows, we consider the evolution of the order parameter which gives the information of the position variable: Before we prove our main theorem, we first classify all stationary solutions to the kinetic equation (2). Proposition 2. The function f e (ω, v) is a stationary solution to (2) if and only if one of the following relations hold: (ii) There exists m 0 ∈ (0, 1) and u ∈ S d−1 such that f e (ω, v) = ((1 − m 0 )δ u (ω) + m 0 δ −u (ω)) ⊗ δ 0 (v).
Proof. We recall Proposition 4.2 in [18] which states that the equilibria of the second-order dynamics (3) with Ω i ≡ 0 coincides with those of the first-order dynamics with Ω i ≡ 0. The results of the particle level can be lifted to the for kinetic level. More precisely, for the first-order kinetic level, it follows from Theorem 1 and Remark 2 in [16] that the stationary solution has the form of f e (ω) = (1 − m)δ u (ω) + mδ −u (ω), m ∈ (0, 1), u ∈ S d−1 .
Hence, we just associate the Dirac measure at zero for the velocity variable to obtain the desired classification of a stationary solution. Now, we prove our last main theorem which shows that a solution to (2) converges to the bi-polar state, except the trivial case.
This implies that the supp v∈R d f 0 has measure zero. Since p t is absolutely continuous, we have supp v∈R d f (t) = 0, t ≥ 0. Hence, we conclude R f (t) ≡ 0, t ≥ 0. (ii) We integrate the energy relation in Lemma 5.3(ii) to get where we represent the potential energy E P in terms of the order parameter as in (10). Our claim is to show that there exists a positive number R * > 0 such that Suppose to the contrary that there exists a positive time t * ∈ (0, ∞] such that lim t→t − * R f (t) = 0.
By letting t → t − * in (13), we have Again the condition (7) yields that the relation (15) implies Since E K is non-negative and continuously differentiable, one has Then, the energy relation (13) gives However, this contradicts the initial assumption (7). Hence, our claim (14) holds.
On the other hand, we use the definition of the order parameter to see Then, it follows from Corollary 1 that We combine (16) and (17) to find and (18) yields the existence of the limit for R f : Thanks to Barbalat's lemma, we also see thatṘ f converges to zero. Finally, our desired assertion follows from the classification of a stationary solution in Proposition 2. This completes the proof.
6. Conclusion. In this paper, we have studied the emergent behavior of the identical second-order swarm sphere model at both particle and kinetic levels. More precisely, we adopt the gradient-like flow approach to show that a solution to identical particle model always converges to the equilibrium. Here, we cannot use the classical theorem as it is due to the presence of the nonlinear centripetal force term and this technical difficulty can be overcome once we follow the proof of the classical theorem step by step. In addition, we establish uniform-in-time 2 -stability with respect to the initial data using the complete aggregation estimate. As discussed before, such uniform-in-time results are rarely found in the literature of the collective dynamics. For the kinetic model, we rigorously derive the meanfield kinetic equation as a by-product of the uniform stability estimate, and the global-in-time existence of a measure-valued solution also directly follows from the measure-theoretic setting. In addition, we investigate the emergent behavior of the kinetic equation by lifting the corresponding results for the particle model. To be more specific, we show that under some initial framework, a solution to the kinetic equation converges to the bi-polar state. In fact, there are still many interesting problems and in particular, emergent dynamics for non-identical problems is largely open. Hence, extension of the presented results to the non-identical case will be pursued in future work.